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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Multiset.v,v 1.1.2.1 2004/07/16 19:31:39 herbelin Exp $ i*)
(* G. Huet 1-9-95 *)
Require Permut.
Set Implicit Arguments.
Section multiset_defs.
Variable A : Set.
Variable eqA : A -> A -> Prop.
Hypothesis Aeq_dec : (x,y:A){(eqA x y)}+{~(eqA x y)}.
Inductive multiset : Set :=
Bag : (A->nat) -> multiset.
Definition EmptyBag := (Bag [a:A]O).
Definition SingletonBag := [a:A]
(Bag [a':A]Cases (Aeq_dec a a') of
(left _) => (S O)
| (right _) => O
end
).
Definition multiplicity : multiset -> A -> nat :=
[m:multiset][a:A]let (f) = m in (f a).
(** multiset equality *)
Definition meq := [m1,m2:multiset]
(a:A)(multiplicity m1 a)=(multiplicity m2 a).
Hints Unfold meq multiplicity.
Lemma meq_refl : (x:multiset)(meq x x).
Proof.
NewDestruct x; Auto.
Qed.
Hints Resolve meq_refl.
Lemma meq_trans : (x,y,z:multiset)(meq x y)->(meq y z)->(meq x z).
Proof.
Unfold meq.
NewDestruct x; NewDestruct y; NewDestruct z.
Intros; Rewrite H; Auto.
Qed.
Lemma meq_sym : (x,y:multiset)(meq x y)->(meq y x).
Proof.
Unfold meq.
NewDestruct x; NewDestruct y; Auto.
Qed.
Hints Immediate meq_sym.
(** multiset union *)
Definition munion := [m1,m2:multiset]
(Bag [a:A](plus (multiplicity m1 a)(multiplicity m2 a))).
Lemma munion_empty_left :
(x:multiset)(meq x (munion EmptyBag x)).
Proof.
Unfold meq; Unfold munion; Simpl; Auto.
Qed.
Hints Resolve munion_empty_left.
Lemma munion_empty_right :
(x:multiset)(meq x (munion x EmptyBag)).
Proof.
Unfold meq; Unfold munion; Simpl; Auto.
Qed.
Require Plus. (* comm. and ass. of plus *)
Lemma munion_comm : (x,y:multiset)(meq (munion x y) (munion y x)).
Proof.
Unfold meq; Unfold multiplicity; Unfold munion.
NewDestruct x; NewDestruct y; Auto with arith.
Qed.
Hints Resolve munion_comm.
Lemma munion_ass :
(x,y,z:multiset)(meq (munion (munion x y) z) (munion x (munion y z))).
Proof.
Unfold meq; Unfold munion; Unfold multiplicity.
NewDestruct x; NewDestruct y; NewDestruct z; Auto with arith.
Qed.
Hints Resolve munion_ass.
Lemma meq_left : (x,y,z:multiset)(meq x y)->(meq (munion x z) (munion y z)).
Proof.
Unfold meq; Unfold munion; Unfold multiplicity.
NewDestruct x; NewDestruct y; NewDestruct z.
Intros; Elim H; Auto with arith.
Qed.
Hints Resolve meq_left.
Lemma meq_right : (x,y,z:multiset)(meq x y)->(meq (munion z x) (munion z y)).
Proof.
Unfold meq; Unfold munion; Unfold multiplicity.
NewDestruct x; NewDestruct y; NewDestruct z.
Intros; Elim H; Auto.
Qed.
Hints Resolve meq_right.
(** Here we should make multiset an abstract datatype, by hiding [Bag],
[munion], [multiplicity]; all further properties are proved abstractly *)
Lemma munion_rotate :
(x,y,z:multiset)(meq (munion x (munion y z)) (munion z (munion x y))).
Proof.
Intros; Apply (op_rotate multiset munion meq); Auto.
Exact meq_trans.
Qed.
Lemma meq_congr : (x,y,z,t:multiset)(meq x y)->(meq z t)->
(meq (munion x z) (munion y t)).
Proof.
Intros; Apply (cong_congr multiset munion meq); Auto.
Exact meq_trans.
Qed.
Lemma munion_perm_left :
(x,y,z:multiset)(meq (munion x (munion y z)) (munion y (munion x z))).
Proof.
Intros; Apply (perm_left multiset munion meq); Auto.
Exact meq_trans.
Qed.
Lemma multiset_twist1 : (x,y,z,t:multiset)
(meq (munion x (munion (munion y z) t)) (munion (munion y (munion x t)) z)).
Proof.
Intros; Apply (twist multiset munion meq); Auto.
Exact meq_trans.
Qed.
Lemma multiset_twist2 : (x,y,z,t:multiset)
(meq (munion x (munion (munion y z) t)) (munion (munion y (munion x z)) t)).
Proof.
Intros; Apply meq_trans with (munion (munion x (munion y z)) t).
Apply meq_sym; Apply munion_ass.
Apply meq_left; Apply munion_perm_left.
Qed.
(** specific for treesort *)
Lemma treesort_twist1 : (x,y,z,t,u:multiset) (meq u (munion y z)) ->
(meq (munion x (munion u t)) (munion (munion y (munion x t)) z)).
Proof.
Intros; Apply meq_trans with (munion x (munion (munion y z) t)).
Apply meq_right; Apply meq_left; Trivial.
Apply multiset_twist1.
Qed.
Lemma treesort_twist2 : (x,y,z,t,u:multiset) (meq u (munion y z)) ->
(meq (munion x (munion u t)) (munion (munion y (munion x z)) t)).
Proof.
Intros; Apply meq_trans with (munion x (munion (munion y z) t)).
Apply meq_right; Apply meq_left; Trivial.
Apply multiset_twist2.
Qed.
(*i theory of minter to do similarly
Require Min.
(* multiset intersection *)
Definition minter := [m1,m2:multiset]
(Bag [a:A](min (multiplicity m1 a)(multiplicity m2 a))).
i*)
End multiset_defs.
Unset Implicit Arguments.
Hints Unfold meq multiplicity : v62 datatypes.
Hints Resolve munion_empty_right munion_comm munion_ass meq_left meq_right munion_empty_left : v62 datatypes.
Hints Immediate meq_sym : v62 datatypes.
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